I have the week off for Thanksgiving, so some friends and I have spent the last few days in Napa hanging out and doing some wine tasting. We got a recommendation to visit Freemark Abbey Winery, which we did, where I came across what seems like a pretty good problem-based investigation. Here is the scenario as they describe it (based on actual events):
"In September 1976 William Jaeger, a member of the partnership that owned Freemark Abbey Winery, had to make a decision: should he harvest the Riesling grapes immediately or leave them on the vines despite the approaching storm? A storm just before the harvest is usually detrimental, often ruining the crop. A warm, light rain, however, will sometimes cause a beneficial mold, botrytis cinerea, to form on the grape skins. The result is a luscious, complex sweet wine, highly valued by connoisseurs."
Basically, they had to decide: harvest the grapes now and guarantee the production of a moderately priced wine, OR wait for the incoming storm and hope that it encourages the mold growth necessary for production of high end wine. There are some accompanying details that the winery used in the decision making process (which I have somewhat shortened here):
  • There was a 50% chance the storm would hit Napa Valley
  • There was a 40% chance that, if the storm did strike, it would lead to the development of botrytis mold
  • If Jaeger pulled grapes before the storm, he could produce wine that would sell for $2.85 per bottle
  • If he didn't pull and the storm DID strike (but didn't produce the mold), he could produce wine that would sell for $2.00 per bottle
  • If he didn't pull and the storm DID strike (and did produce the mold), he could produce wine that would sell for $8.00 per bottle (although at 30% less volume because of the process to produce this wine)
  • If he didn't pull and the storm DIDN'T strike, he could produce wine that would sell for $3.50 per bottle

I can imagine several different versions of the problem depending on the group of students you were working with (by adding/removing details and alternative scenarios). I am attaching the printouts that I photographed in case anyone wants the details.
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The 12th graders and I started a new unit/project/problem/whatever last week. It started with this image:

What is this?

We took some guesses before someone correctly identified the image as the readout from a stationary bike machine. If this was the readout, I nudged, then:

What did the course look like?

I let them play with it for a while. They identified some important questions they had and information they wanted:
  • What do the dots represent?
  • How can we quantify this?
  • Does speed matter?
  • Did the bike have gears?
  • What do we mean when we say, "what does the course look like?
Ultimately, they agreed upon some answers and, as a result, gave themselves some space to work within. This part takes time and I really think we need to give it the time it deserves. If we don't let students pose and answer those questions, the rest of the unit suffers. I let them know that we would examine some simpler cases first and return to the unit question somewhere down the road.

So, the next day they came back and and I had them map the course for these two:
This took another class period for them to sort out. By the end, though, they started to feel pretty good about the fact that the dots were a measure of resistance which they connected to the slope of the course at that point. Towards the end of class, someone asked, "What would it look like if you were going downhill?' They talked for a bit and devised a system to handle that, so the next day I gave them this:
This allowed them to have conversations about what it meant when the resistance was positive, what it means when resistance is zero, and what it means when resistance is negative.

It's fun to watch the students construct an understanding of antiderivatives. The three-day span reminded me of a few things:
  • the importance of listening to how students are thinking about and making sense of problems
  • our role as teachers in responding to that way of knowing by bringing what might be the "next good problem" for them
  • allowing them the time to sort things out together

I feel like there are a couple ways to go with this trajectory next:
  1. Stay with resistance, but use a continuous curve
  2. Stay with this discrete model, but switch contexts (maybe speed or rate of growth)
  3. Stay with discrete and resistance, but use a more complicated readout

I'm leaning towards #1. What do you think would be best for the students?
Our Juniors are away on Internship right now so I have been driving all over the city checking in on them and seeing what they are up to. With all that time in the car I was doing some serious project brainstorming for next year. I wanted to put my ideas down in a place where I could come back to them (especially because some of these are still real rough drafts). I'm sticking to the problem-based format because I like it...a LOT. In each of these, students would have to create the mathematics as it emerges in their exploration with the problem.  I would love some blogger critique so please let me know what you think...

HORIZON: How far away is the horizon?

This is a pretty famous problem in mathematics (I think). I remember seeing it and being intrigued. I think there is an opportunity for pretty rich math here. Students would have to create an abstraction to represent the situation, develop rules about their abstraction, and use them to approximate an answer to the unit question.

PROBLEMATIC PACKAGING: How can you optimize this packaging?

This one arose out of a curiosity I had about optimizing parking lot design. I thought it might be fun as a puzzle where students get a package and different "items" (blocks of two or three sizes and shapes and worth various points) and they have to figure out how to maximize their point value. Could be extended by looking at different point values or different size boxes.

GET ME OUTTA HERE: How do you know if a game is solvable?

I love these puzzles. I'm not exactly sure how to turn this into a full-blown unit, but I'm pretty sure it can be done. This might not be the best unit question. I have also considered giving students a specific puzzle and asking "What is the fewest number of moves?" Maybe we can extend it from there and move into "solvable" setups?

7 CLICKS FROM KEVIN BACON: What is the minimum number of clicks to get to Kevin Bacon from ANY person on Facebook?

This one I'm REALLY not sure about. Obviously, it comes from the popular game about social connection (although it might be a good idea to use a celebrity my student will have actually hear of). I thought there might be some connections to graph theory here and it might be a nice extension of a combinatorics unit that I will do again next year ("How many combinations are there at Chipotle?"). Feedback please.

WIN, LOSE, or DRAW: Can you draw this without lifting your pencil or going over the same line twice?

The Bridges of Konigsberg is such a GREAT problem that I want to turn it into a whole unit next year. I'm not sure I love the context of the original problem because, to students, I think it feels like a pseudocontext. I thought about giving them a crazy network diagram and asking the unit question about that (maybe with "bridges" showing up along the way?). However, I usually prefer to start with a concrete situation and abstract from there....not sure.

...something more complicated but this was the best picture I could find for now

GOING COASTAL!: How long is the California coastline?

I did a similar project this year where students figured out the area of the Koch Snowflake. I'm thinking about trying to start concrete and move abstract next year with this one. I'm sure the Koch Snowflake will still rear it's ugly head somewhere in our investigation.
I'm getting ready to plan a problem-based unit on probability and I'm looking for a unit question that can drive our entire investigation. So, this morning I spent about an hour looking online and I was shocked by what I did NOT find. Probability has to be one of the most interesting an relevant topics in high school math and there was not much that caught my attention (and if it doesn't catch my attention, I'm pretty sure it won't catch the students' attention). Here are possible questions that I'm considering right now:

1. The Birthday Problem
What is the probability that, in a room with "n" people, at least two will have the same birthday.

2. Probability Game
I don't know of a great one, but I was thinking about using a "game" to drive our investigation (something like Pig...but preferably more complex)

3. Sports
What is the probability that Team X will make it to the NBA Finals (or something along those lines)

4. Insurance Pricing
Is insurance pricing fair?

The only thing that I really DON'T want to do is Casino/Gambling stuff. Other than that, any suggestion for a unit question is fair game...

p.s. I'm really tempted to create a tab where "we" can start to organize unit questions for reference. Good idea?